Markov chain - stationary distribution - please help

-A taxi is located either at the airport or in the city. From the city, the next trip is to the airport with 1/4 probability, or to somewhere else in the city with 3/4. From the airport, the next trip is always to the city.

(a) find the stationary distribution
(b) starting from the airport, what is the expected number of trips until its next visit to the airport?

2. Relevant equations

3. The attempt at a solution

The matrix I got is

0 1
1/4 3/4

but I am having trouble solving for the stationary distribution and expected number of trips. Somehow, I don't think the stationary distribution even exists. But it should. What am I missing? Thanks.

your transition matrix is same as expected.And the stationary distribution exists. you can find the stationary distribution by solving it as two linear equations.
let 'Pij' be the element, i=1,2 and j=1,2 (for the given transition matrix).

P0=P0*P00+P1*P01; P1=P0*P10+P1*P11

here P0 and P1 are unknowns , the elements of stationary distribution.
Pij(P00,P01,P10,P11) are the transition matrix elements.